Calvin, W. J., Smith, B. D., A.I.Ch.E. J. 17, 191 (1971). Chen, N. H., J. Chem. Eng. Data 10, 207 (1965). Francis, P. G. McGlashan, M. L., Hamann, S. D., McManamey, W. J., J. Chem. Phys. 20, 1341 (1952). Fredenslund, A,, Sather, G. A., IND.ENG.CHEM.,FUNDAM. 8,
Rossini, F. D., Ed., “Selected Values of Properties of Hydrocarbons and Related Compounds,” American Petroleum Research Project 44, Carnegie Press, Pittsburgh, Pa., 1953. Scatchard, G., Wood, S. E., Mochel, J. M., J . Amer. Chem. SOC.
Fu, S. J., Lu, B. C. Y., J. Appl. Chem. 16, 324 (1966). Glaser, F., Ruland, H., Chern.-Zng.-Tech. 29, 772 (1957). Goates, J. R., Sullivan, R. J., Ott, J. F., J . Phys. Chem. 63, 589
Scatchard, G., Wood, S. E., Mochel, J. M., J . Amer. Chem. SOC.
718 (1969).
(1959).
Grosse-Wortman, H., Jost, W., Wagner, H. G., 2. Phys. Chem., Frankfurt Am. Main 49(1-2), 74 (1966). Houng, J. J., D.Sc. Dissertation, Washington University, St. Louis, -Missouri, 1969. Houng, J. J., Smith, B. D., A.I.Ch.E. J . 17, 1102 (1971). Jordan, T. E., “Vapor Pressures of Organic Compounds,’’ Interscience Publishers, New York, N. Y., 1954. Killian, H., Bittrich, H. J., 2. Phys. Chem., L e i p i g 230(5-6), 383 (1965).
Kortum, G., Freier, H. J., Chem.-Ing.-Tech. 26, 670 (1954). Kozicki, W., Sage, B. H., Chem. Eng. Sci. 15, 270 (1961). Leach, J. W., Ph.D. Dissertation, Rice University, Houston, Texas, 1967. Leland, T. W., Chappelear, P. S., Gamson, B. W., A.1.Ch.E. J . 8, 482 (1962).
Leland, T. W., Rowlinson, J. S., Sather, G. A., Watson, I. D., Trans. Faradau SOC.65. 2034 (1969). Longuet-HigginG H. C., h o c . Roy. Soh., Ser. A 205, 247 (1951). Lu, B. C. Y., Jones, €1. K. D., Can. J . Chem. Eng. 44,251 (1966). Lundberg, G. W., J. Chem. Eng. Data 9, 193 (1964). McGIashan, M. L., Potter, D. J. B., Proc. Roy. SOC.,Ser. A 267, 478 (1962).
Mrazek, R. V., Van Ness, H. C., A.I.Ch.E. J. 7, 190 (1961). Nicholson, D. E., J. Chem. Eng. Data 6, 5 (1961). Noordtzig, R. M. A., Helv. Chim. Acta 39,637 (1956). Palmer, D. A., D.Sc. Dissertation, Washington University, St. Louis. Missouri. 1971. Pitzer, K. S., J. Chem. Phys. 7,583 (1939). Powell, R. J., Swinton, F. L., J . Chem. Eng. Dota 13, 260 (1968). Rao, V. N. K., Swami, D. R., Rao, ?VI.N., A.I.Ch.E. J. 3, 191
61, 3206 (1939a).
Scatchard, G., Wood, S. E., Mochel, J. AI., J. Phys. Chem. 43, 119 (193913).
62, 712 (1940).
Schulze, A., 2. Phys. Chem., Leipzig 86, 309 (1914). Smith, B. D., Holt, D. L., Stookey, D. J., Yuan, I-C., “Evaluation of Thermodynamic Excess Property Data for Nonelectrolyte Mixtures-Subcritical, Miscible, Binary, Mixtures of Hydrocarbons with Five to Eight Carbon Atoms,” Thermodynamics Research Laboratory, Washington University, St. Louis. Missouri. 1971. Stookey, D’. J., Smith, B. D., paper presented a t the Houston Meeting of the A.1.Ch.E. in March 1971. Stull, D. R., Ind. Eng. Chem. 39, 517 (1947). Timmermans, J., Ed., “Physico-Chemical Constants of Pure Organic Compounds,” Vol. 1, Elsevier Publishing Co., New York, N.Y., 1950. Van Ness, H. C., “Classical Thermodynamics of Non-Electrolyte Solutions,” Pergamon Press, New York, N. Y., 1964, p 79. Vilcu, R., Stanciu, F., Rev. Roum. Chim. 1 1 , 175 (1966). Waddington, G., et al., J . Amer. Chem. SOC.69, 27 (1947). Washburn, E., Ed., “International Critical Tables,’’ McGrawHill, New York, N. Y., 1926. Weast, R. C., Ed., “Handbook of Chemistry and Physics,” 47th ed., The Chemical Rubber Co., 1966. Werner, V. G., Schuberth, H., J . Prakt. Chem. 31, 225 (1966). Wheeler, J. D., Ph.D. Dissertation, Purdue University, Lafayette, Indiana, 1964. Wheeler, J. D., Smith, B. D., A.Z.Ch.E. J . 13, 303 (1967). Willingham, C. B., Taylor, W. J., Pignocco, J. &I.,Rossini, F. D,, J . Res. S a t . Bur. Stand. 35, 219 (1945). Young, S., Sci. Proc. Roy. Dublin SOC.,S.S . 12, 374 (1910). Yuan, I-C., D.Sc. Dissertation, Washington University, St. Louis, Missouri, 1971.
,.
RECEIVED for review October 1, 1971 ACCEPTED June 2, 1972
(1067) \A””.
Reamer, H. H., Sage, B. H., J . Chem. Eng. Data2, 9 (1957).
Correlation of a Partially Miscible Ternary System with Conformal Solution Theory David A. Palmer,l Wallace I.-C. Yuan,* and Buford D. Smith* Thermodynamics Research Laboratory, Washington University, St. Louis, Mo. 63230
+
+
Extensive GE,HE, and VE data for the partially miscible acetonitrile benzene n-heptane system have been correlated successfully with a formalism based on conformal solution theory. The GE correlation was superior to that obtained with the three-parameter NRTL equation. The general weighting function approach suggested in 1964 b y Wheeler was used. The radius of gyration was used to represent size-shape effects. Pair weighting parameters were determined for those molecular pairs with identifiable special interactions. A general correlation based on this approach may be feasible for mixtures containing polar molecules.
A
previous paper (Yuan, et a ~1972) , described a correlation structure for hydrocarbon (nonpolar) systems based on the conformal solution formalism. Two binary constants, obtained from GE and VE data at one temperature, were required to correlate the binary GE and VE data a t one temperature. 1
111.
Present address, Amoco Chemicals Corporation, Naperville,
Present address, Chemical Engineering Department, Tunghai University, Republic of China.
Four binary constants, obtained from either GE and VE data or H E and VE data a t two temperatures, were required to predict GE, HE, and VE data as a function of both composition and temperature. The correlation utilized the Leland, et ai. (1962), equations to relate the mixture conformal parameters to the like- and unlike-pair parameters. The present paper extends the correlation structure to the partially miscible acetonitrile (1) benzene (2) n-heptane (3) system which includes a polar molecule. (The numbers
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Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
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after the component names will be used to identify the various pair parameters.) Extensive GE, HE, and VE data for the binaries and GE data for the ternary have been measured by Palmer (1971). The results indicate that the correlation package now being developed for the hydrocarbon and simple fluid systems can eventually be extended to include polar molecules. Application of Corresponding-States Equations
Pure-Component Conformal Parameters. The previous paper, which hereafter will be denoted by YPS, described the calculation of the pure-component conformal parameters, faa and ha,, from the configurational Gibbs free energies, eq YPS-5, and volumes, eq YPS-6, of component “a” and the reference substance “0.” It also described how the validity of the corresponding states equations can be checked by using and ha, values thus obtained to predict the configurathe jaa tional enthalpy of component “a” using eq YPS-3b. The predicted configurational enthalpy values for acetonitrile and n-heptane a t 45”C, using benzene as the reference, differed by 0.68 and 0.43%, respectively, from the values calculated from heat of vaporization and gas nonideality data. These predictions are well within the experimental error of the heat of vaporization data. Acetonitrile is not generally considered to be in the same corresponding states class as benzene and n-heptane. The critical compressibility factor calculated from the handbook values for the critical properties of acetonitrile is 0.181, compared to the benzene and n-heptane values of 0.267 and 0.259. Xevertheless, the accurate prediction of the configurational enthalpies means that the configurational free energy, enthalpy, and volume surfaces for the three components have been made to “correspond” through use of the temperature and pressure reducing parameters, ”faa and jaa/haa, All those factors (such as nonsphericity, noncentral forces, attractive forces other than the van der Waals-London type, etc.) which tend to make dissimilar molecules depart from the simple twoparameter corresponding-states principle are evidently taken into account when the faa and ha, parameters are allowed to vary as necessary with temperature (and pressure) to bring the configurational free energy and volume surfaces for “a” and “0” together. If so, there is a possibility that even hydrogen-bonded molecules can be made to correspond by this procedure. The calculation of the necessary faa and ha, values requires only that vapor pressure and volume data be available for the pure components over the temperature range of interest. Mixture Conformal Parameters. The mixture con. formal parameters for each binary a t 45°C were calculated from Palmer’s (1971) binary GE and VE data by iteration with the following equations.
binary a t 20°C (Werner and Schuberth, 1966), and 80°C (Brown, 1952) were also used in the correlation work described below. Correlation of Mixture Conformal Parameters
The random equations which worked well for the simple fluids and the Leland equations which worked well for the Ce and C7 hydrocarbons in the YPS paper both failed on the acetonitrile systems. In the absence of any satisfactory theoretr ical averaging equations, it was necessary to resort to the empirical weighting function approach suggested by Wheeler and Smith (1967). General Weighting Functions. The correlation equations used for the mixture conformal parameters were as follows.
The S’s denote size-shape factors and the I’s denote special interaction factors. The primes in the kXequation indicate that the size-shape and the interaction effects will be weighted differently in the two equations. Of all of the various mixture parameters in eq 3 and 4, the unlike pair parameters, j a b and hab, have the largest impact on the predicted results. These are the primary determinants of the magnitude of the predicted excess property value. Very small changes in the fab and hab values cause large changes in the predicted results. The weighting factors, Sa and Z(a,b), are the primary determinants of the shape of the predicted excess property curves. They determine how flat the curve is a t the top, how skewed it is, and can even make it S-shaped when the predicted excess property values are numerically close to zero. Changes in the weighting factors do not affect the shape of the curve as strongly as proportional changes in jab or hab affect the magnitude. This means that the weighting factors need not be known as accurately as the f,b and hab values. Identifiable Bond Model. Two general approaches t o the specification of the S and I values have been described by Palmer (1971). Both approaches obtained the S values as described below. One approach related the I parameters to pure component properties; further work will be needed to determine its ultimate value. The other approach, which will be presented here, was based on an “identifiable bond” model. The identifiable bond model postulates that the I(a,b) factor will be nonunity only if there is some identifiable reason for a special interaction between molecules “a” and “b” beyond the van der Waals-London forces. The identifiable special interactions present in the acetonitrile (1) benzene (2) f n-heptane (3) system include: (a) the interaction between two C=N bonds; (b) the interaction of ?r bonds in two benzene rings; (c) the interaction of a benzene ring with the C r N group; and (d) interaction of n-heptane with the C r N group due to an induced dipole in the n-heptane molecule. The heptane-heptane and heptane-benzene interactions are assumed to be of the van der Waals-London type and are assigned unity I values. The identifiable bond model will be of general use only if the Z values determined from mixture data on selected systems are transferable to other systems where the same interactions toluene occur. Data are being obtained on the acetonitrile
+
Temperature Dependence. The temperature dependence of the mixture conformal parameters for the acetonitrile (1) benzene (2) and the acetonitrile (1) mheptane (3) systems could not be fully explored because of the lack of sufficient data a t a temperature other than 45OC. The temperature dependence in the benzene ( 2 ) n-heptane (3)system has been described in the YPS paper. Data on the latter
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Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
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n-heptane system a t 45°C to test whether the I values obtained below from Palmer’s (1971) data can be used when toluene ieplaces benzene in the mixture. The identifiable bond model may be particularly attractive for the correlation of hydrocarbon systems. The only a,b pairs with nonunity I(a,b) values mill be those in which each molecule either contains one or more double bonds or is an aromatic molecule. Determination of the S and I Values. It is postulated that the S , and S’, parameters should reflect only size and shape effects, should be independent of temperature over the correlation range (or a t least be weak functions of temperature), and should be expressed in terms of molecular properties readily available in handbooks or other compilations. The first and second criteria eliminate the molar volume; that property includes the effect of intermolecular bonding and it varies rapidly with temperature. The two properties investigated in this study were the molecular weight and the radius of gyration. The latter quantity has been tabulated by Thompson (1966) for 550 compounds. It is a simultaneous size-shape parameter in that it reflects not only the size but the way the mass is distributed about the center of gravity of the molecule. The numerical values used for the S parameters do, of course, affect the I values necessary to reproduce the ‘‘experimental” fn and h, values calculated from the experimental binary GE and VE datum points with eq 1 and 2. Intuitively it is felt that the radius of gyration should be a better sizeshape factor than the molecular weight which reflects only size. Another reason for selecting the radius of gyraticm (RG)is illustrated in Figure 1. That figure also illustrates the way the various nonunity I(a,b) values were determined. The root-mean-squared-deviation ( R M S D ) in fz3 plotted as the ordinate in Figure 1 is a measure of how the calculated f8b values varied with composition for any given pair of assumed S and 1(2,2) values. (The 1(2,3) and 1(3,3) values are unity in the identifiable bond model.) The unlike-pair parameter jab must be essentially independent of composition if one average value is to reproduce accurately the experimental GE and VE data across the composition range, and this fact was used as a criterion in the choice of the best S and Z values. The R X S D in j a b was based on the deviations of the individual calculated jab values from the average of all values across the composition range. The five curves in Figure 1 represent five different assumptions for the S factors. All gave an acceptable R M S D in fz3; i.e., the average f23 value in each case reproduced all of the experimental GE and VE points within experimental accuracy. However, it is reasonable to expect the benzene-benzene I value to lie fairly close to 1.0 and it was arbitrarily assumed a t this point that the weighting given a pair with special interaction should be greater than the unity weight given to pairs with no special interactions. The 1(2,2) value in Figure 1 is greater than unity only when the radii of gyration for benzene and n-heptane are substituted for SZand Sain eq 3. The best 1(2,2) value when the radii of gyration are used is 1.25, and that is the value shown in Table I where all of the parameters are listed. A plot for h23 similar to Figure 1 showed that S’Z = SI3 = Z‘(2,3) = 1.0 worked as well as other combinations. The unity weighting for all of the molecules and pairs in eq 4 agrees with the Leland equation for kX. Once all of the Sa values are fixed a t the radius of gyration values, and all of the S’, values are set equal to unity, the Z(a,b) and I’(a,b) values in the other binaries can be deter-
“-t n
BENZENE 12) + n-HEPTANE 131 AT 45°C
I
I
8
1.0
I
.‘I
.2
0
.6
iwi
Figure 1 . Effect of different assumptions for the factors on the numerical values of /(2,2)
Table
I
I
1.2
1.4
Sz
and Sa
I. Weighting Function Parameters at 45°C
Component
Sa
Acetonitrile (1) Benzene (2) %-Heptane (3)
1.821 3.004 4.267
I(a,a)
S ’a
5.12 1.25 1.00
1.0 1.o 1.0
I‘(a,a)
1.0 1.0 1.0
System
\(ab)
I‘(o,b)
Acetonitrile (1) benzene (2) Acetonitrile (1) n-heptane (3) Benzene (2) n-heptane (3)
3.56 2.90 1.oo
1.048 1.044 1,000
+
+ +
mined by a two-variable iteration analogous to that illustrated benzene (2) system in Figure 1. For the acetonitrile (1) with 1(2,2) = 1.25, the lowest R M S D in f1z occurred a t I(1,l) = 4.175 and 1(1,2) = 2.75. However, any combination of I ( l , l ) , I ( l , 2 ) from 3.2,2.0 to 5.6,4.0 reproduced the experimental GE and VE data within experimental error. For the acetonitrile (1) n-heptane (3) system, the lowest R M S D in f13 occurred a t 1(1,1) = 5.12 and 1(1,3) = 2.90. Again, there was a range of I(1,1),1(1,3)pairs from 4.9,2.6 to 5.4,3.4 which reproduced the experimental GE and VE data within experimental accuracy. The range of permissible parameter values for this partially miscible binary was much narrower than for the two miscible binaries. Also, the I(l,l)= 4.175 value associated with the minimum R V S D in f12 for the acetonitrile (1) benzene (2) system did not fall within the permissible range of I(1,l)values for the partially miscible acetonitrile (1) n-heptane (3) system. However, the I(1,l) = 5.12 value which gave the best fit for the latter system also fits the acetonitrile (1) benzene (3) very well. Consequently, the final choice for I(1,l) was 5.12. The corresponding 1(1,2) and 1(1,3) values were then 3.56 and 2.90, respectively. The final correlation values are listed in Table I. The R M S D in hlz for the acetonitrile (1) benzene (2) system remained constant a t a very low value as the Z ’ ( l , l ) , I’(1,2) pair was varied in small increments from 1.0,1.048 to 1.035,1.064. TheI‘(1,l) = l.OandI’(l,2) = 1.048 set was arbitrarily picked. With I’(1,l)= 1.0, the Z’(1,3) value obtained from the acetonitrile (1) n-heptane (2) data was 1.044. All correlating equations will have a range of permissible parameter values. The size of the range will depend upon (a) the criterion used to define an acceptable fit of the data, and (b) the difficulty of fitting the data. The criterion used here for an acceptable fit was that the experimental data must be fitted within the experimental accuracy. This permitted a
+
+
+ +
+
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Ind. Eng. Chem. Fundam., Val. 1 1 , No. 3, 1972
399
~
I
Acetonitrile (1) t Benzene (2)
I
Acetonitrile (1) + Benzene (2) at 45°C
+. 1 5 1
20°C Werner, Schuberth (1966)
3.2
+.
3.0
.1
.2
.3
.4
.5 .6 x1
.7
.8
0 45OC Palmer (1971)
tL
d I
.9
Figure 2. Prediction of Palmer’s experimental VE data (1 971) for acetonitrile benzene with three different
+
models
I
Table II. Binary Constants System
t,
+
Acetonitrile ( 1 ) benzene (2) Acetonitrile (1) n-heptane (3) Benzene (2) n-heptane (3) Benzene (2) n-heptane (3) Benzene (2) n-heptane (3)
+ + +
+
O C
45 45 20 45 80
kab
0 043737 0.130097 0.032838 0.030504 0.027937
Cab
-0.001596 -0.017104 -0.006684 -0.006271 -0.005775
rather wide range of permissible parameter values for the relatively easy-to-fit miscible binaries. The range of permissible values for the difficult-to-fit partially miscible binary was much narrower. No great significance should be attached to that pair of parameter values which best fits the data for any one binary. The exact location of the minimum deviation in the parameter space will always be affected by errors in the experimental data, by numerical “static” which invariably arises in the iterative numerical calculations, and by inadequacies in the correlation equations themselves. The “best” I(a,b) value will be that one which works for the maximum number of binaries in which the a,b interaction occurs. Binary Constants. The YPS paper defines the k a b and cab as the binary constants where
Table 11 gives the values resulting from the fab and hab values obtained from the S and Z values in Table I. The typical variations of kab and Cab with temperature are illustrated by n-heptane values; the values listed differ the benzene slightly from those given in the previous paper because of the inclusion of the new data a t 45OC. As explained in the YPS paper, the equations
+
1 - klz = A N e x p ( B d T )
(7)
1-
(8)
CIZ
=
C~zexp(DdT)
appear to be suitable correlations for the effect of temperature 400 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
I
I
I
.1
.2
.3
I
I
I
I
I
I
.4
.5
.6
.7
.8
.9
Figure 3. Prediction of activity coefficient data for the benzene system at 45°C acetonitrile
+
on the kab and cab values. The values of the constants obtained n-heptane system were by Palmer (1971) for the benzene A = 0.995713, B = -8.489270, C = 1.001279, and D = 1.582739. These are based on the 45 and 80°C data and differ slightly from the YPS values which were based on the 20 and 80°C data. The 20°C data are the least accurate of the three sets. These A , B , C, D values are based on the 1(2,2) value a t 45°C shown in Table I. Palmer (1971) calculated 1(2,2) values from data a t 20, 45, 75, and 8OOC. There was no correlation with temperature and it was concluded that the differences in the four 1(2,2) values obtained reflected differences in quality of the various data sets more than the effect of temperature. The results from the best two sets of data (45 and 8OoC) agreed quite closely. Consequently, until the I parameters can be investigated over a wider temperature range with good data, it will be assumed that the interaction parameters are independent of temperature.
+
Reproduction of Binary Data
The reproduction of Palmer’s VE data (1971) is shown in Figure 2 where the results from the identifiable bond model are compared with the random form and the Leland equation results. This gives a good illustration of the inability of the Leland and random forms to work for acetonitrile systems. Figure 3 shows the prediction of acetonitrile benzene activity coefficientsa t 45°C. It should be remembered that the correlation constants were obtained from GE and VE data. The activity coefficient equations must be obtained by differentiation of the GE equation with respect to composition, and prediction of the activity coefficients with constants obtained from GE data is a sensitive test of the correlation accuracy. The prediction of the benzene n-heptane activity coefficients is shown in Figure 4. The correlation constants were obtained from GE and VE data a t 45 and 8OOC. The predictions a t those temperatures are good. The agreement between
+
+
Acetonitrile (1) + n-Heptane (2) at 45°C
Benzene (2) t n-Heptane (3)
d fI
0 20°C Werner, Schuberth (1966)
2.0
0 45°C Palmer (1971) 0 80°C Brown (1952)
*J . 9
0 20°C Werner, Schuberth (1966) 0 45°C Palmer (1971)
< 1.8
.-u
0
0)
om
1.7
0
sa 1.6
.-.-> 1.5
0
73
I
1.4 1.3 1.2
1.1 1.0 I
I
1
I
I
.1
.2
.3
.4
.5 x2
Figure 4. Predictibn of the benzene coefficients at three temperatures
260
I
.6
1
.7
I
.8
A
I
.02
.9
+ n-heptane activity
ACETONITRILE (1) t n-HEPTANE (3) AT45”C
.04
.[)6
.08
.10 x1
.94
.96 .98
Figure 6. Prediction of the activity coefficients for the acetonitrile n-heptane system at 45OC
+
The reproduction of the GE and VE data for the partially miscible binary was very good, as shown in Figure 5. The prediction of the activity coefficients shown in Figure 6 was not as good but was much better than the results obtained with the Van Laar equation when the Van Laar constants were obtained from the GE data. Multicomponent Equations
The multicomponent activity coefficient equation is
R T In
Figure 5. Reproduction of the correlated GE and VE data for the partially miscible binary. The data are from Palmer
(1 9711
the predicted and experimental values is not so good for the 2OoC data a t low benzene concentrations. This discrepancy may be partially due to the fact that the 2OoC data do not agree as well with the Gibbs-Duhem and Gibbs-Helmholtz equations as do the sets a t 45 and 80°C.
T~ =
G”p(T,p,z) - G”j(T,p)
+
The partial derivatives of fn and h, are obtained by differentiation of eq 3 and 4, respectively. For systems a t low pressures, the pV terms are negligible and the configurational properties can be considered to be functions of temperature only. Equation 9 reduces to the Leland equation binary form, eq YPS-15, when the following weighting factors are used: Z(a,b) = hab, Z’(a,b) = 1.0,8, = 1.0, andh”, = 1.0. Ternary Predictions
The accuracy of the predictions for Palmer’s ternary GE values (1971) is illustrated in Figure 7. Nine of the 51 ternary points were along the solubility envelope and the rest were scattered throughout the miscible region of the acetonitrile benzene n-heptane ternary. The prediction utilized the weighting parameters from Table I and the constants from
+
+
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
401
ACETONITRILE (1) t BENZENE 12) t n-HEPTANE (3) AT45'C 400 ACETONITRILE + BENZENE t n-HEPTANE AT 45°C
f 300
.-cE
2
3200 W
0
100
100
200 300 GE (Predicted) ~
400
.2
Figure 7. Comparison of the predicted and experimental ternary GE values. The experimental data are from Palmer
(1 9711 -Heptane
Table 11, all of which were obtained from binary data only The root-mean-square-deviation for the 51 predicted values was 9.6 cal/mole, an error which is approximately 3% of the average GE value. The predicted vapor concentrations were within 0.02 mole fraction 92% of the time. The average deviations calculated from
Ayi =
$ ~
IAYi i j
N
where N refers to the number of experimental points = 0.0117, y2 = 0.0055, and = 0.0097. The were average pressure deviation was 0.0076 atm. A comparison was made with the NRTL equation (Renon and Prausnitz, 1968). The three NRTL parameters were found for each binary system by fitting the GE data. The prediction of ternary GE values was not as good as with the conformal solution equations, the RMSD being 25.7 cal/mole compared to 9.6 cal/mole. However, in this single application the identifiable bond model had more adjustable parameters than the NRTL equation. The conformal solution approach will be superior only if the S and I values used are general and can be applied to other systems. Ternary Solubility Envelope. A correlation for activity coefficients can be used to predict the ternary solubility envelope and tie lines with the method of Null (1970). Such predictions are difficult to make because very small errors in the predicted activities result in a large displacement of the solubility envelope on the composition grid. This happens because the activity surfaces are quite flat near the plait point. The prediction a t 45'C is shown in Figure 8. The prediction a t 20°C was similar in that the predicted solubility envelope was too high in the region of the plait point. The mislocation of the solubility envelope is due primarily to the 1(1,3) parameter, which is the one most difficult to determine for a partially miscible system.
*1
Acetonitrile
Figure 8. Prediction of the solubility envelope from binary data only
possible the correlation of all three excess properties of a partially miscible ternary system. The correlation was good but not perfect. It was better than those obtained with other equations such as the NRTL equation and obviously could have been improved by further refinement of the binary weighting parameters. This was not done because the parameter values will have little significance unless they can be transferred to other systems where the same pair interactions occur. Determination of their best values must await a broader investigation covering many binary and ternary systems. After the best S, and I(a,b) values have been established for a wide range of molecules and pairs, it will probably still be necessary to determine the binary constants k12 and CIZ from mixture data. It will be a great help if these values can be obtained from a limited amount of easily measured HE and V E data. Both HE and V Edata will be necessary because past experience with many correlation forms has indicated that neither H E nor VE data alone provide enough experimental information to permit the accurate prediction of the VLE data. The conformal solution theory has been shown to be capable of simultaneously correlating all three excess properties for a wide variety of systems and has been shown many times to be capable of accurately predicting the HE data with constants obtained from GE and VE data. Stookey and Smith (1971) has recently developed an algorithm which accurately predicts the binary VLE data as a function of composition and temperature from HE and VE data a t two convenient temperatures. These developments should make it feasible to develop a general correlation for mixtures of polar molecules even in the absence of a valid theoretical equation for the mixture conformal parameters. Acknowledgment
Discussion
The conformal solution formalism (as described in the YPS paper) plus the use of the general weighting function approach proposed earlier by Wheeler and Smith (1967) have made 402 Ind. Eng. Cham. Fundam., Vol. 1 1 , No. 3, 1972
This research was supported by the American Oil Foundation Design Fellowship held by Dr. Palmer, by National Science Foundation Grant GK-1971, and by the Industrial Participants in the Thermodynamics Research Laboratory.
The assistance of the Washington LTIiiversity Computing Facility through National Science Foundation Grant G22296 is gratefully acknowledged. Nomenclature
See the previous paper by Yuan, et al. (1972). literature Cited
Brown, I., Aust. J . Sci. Res., Ser. A 5, 530 (1952). Leland, T. W., Chappelear, P. S., Gamson, B. W., A.I.Ch.E. J . 8 , 482 (1962).
Null, H. It., “Phase Equilibrium in Process Design,” Wiley, New York, N. Y., 1970. palmer, D, A., D.+. ~ i ~ ~ Washington ~ ~ t ~ University, t i ~ ~st., Louis, Missouri, 1971. Renon. H.. Prausnitz. J. R‘I.. A.I.Ch.E. J . 14, 135 (1968). Stookey, D. J., Smith, B. D., paper presenied at‘the Houston meeting of the A.1.Ch.E. in March 1971. Thompson, W. H., Ph.D. Dissertation, Pennsylvania State University, University Park, Pa., 1966. Werner, G., Schuberth, H., J . Prakt. Chem. 31(5-6), 225 (1966). Wheeler. J. D.. Smith. R. D.. A.I.Ch.E. J . 13. 303 (1967). Yuan, W. LC:, Palmer, D. A., Smith, B. D.,‘IND.ENQ.’CHEM., FUNDAM. 11, 387 (1972). RECEIVED for Review October 1, 1971 ACCEPTED June 2, 1972
A Root-Locus interpretation of Modal Control B. R. Howarth,’ E. A. Grens 11,” and A, S. Foss Department oj Chemical Engineering, University of California, Berkeley, Calij. 94730
Modal control for single modes of linear, lumped-parameter processes i s examined from the point of view of root-locus methods. Such control is equivalent to exact pole-zero cancellation. Sensitivity of the controlled process to parameter variations may therefore be high. However, this i s a significant effect only when the gain i s chosen such that ihe closed-loop eigenvalue of the controlled mode i s nearly equal to that of some other mode. At other gain values the behavior of the single mode controller i s quite insensitive to large perturbations in controller parameters. Such observations suggest that control systems with restricted measurement and manipulation may achieve results for control of a single mode approaching those available with an ideal modal controller.
T h e concept of modal control for chemical processes seems to have been first advanced by Rosenbrock (1962). In its original form, this approach is very attractive from a theoretical viewpoint. The reduction of overall process response times by action on the decay times of the modes and the noninteraction of the control loops, together with the simplicity of the control law, have tended to overshadow the practical implications of the large number of measured and manipulated variables required for implementation. While these requirements for measurement and manipulation restrict the possibilities of application of modal control, nevertheless a better understanding of the effects of modal control may assist the design of more practicable controllers based on the principles of modal analysis. By consideration of the special case of control of only one mode, we have found that the behavior of modal controllers can be interpreted in terms of root-locus theory. This interpretation derives from an analysis of the structure of the characteristic equation of the closed-loop system; it is based on the observation that modal controllers achieve their effect through pole-zero cancellation. Thus a high sensitivity of these controllers to variations or uncertainties in their parameters is to be expected and can be investigated through root-locus techniques. It is possible to extend this treatment of single-mode control to the control of several modes in a qualitative way, but the conclusions that can be drawn from the more general case are not so clear. Present address, Honeywell Pty. Limited, Sydney, Australia.
State Matrix for the Closed-Loop System
For small deviations from some steady state, the response of many chemical processes can be represented by linear, lumped-parameter models. Then the process dynamics can be represented by the so-called state equations
X=A.X+Bu
y
=
cx
(1)
(2)
where A is the constant state matrix for the uncontrolled system, B is the constant control-effect matrix, C is the output matrix, u is the control vector (dimension m), x is the state vector dimension n), and y is the output vector (dimension T ) . Here the state matrix A is assumed to have distinct eigenvalues XI, X2, . , ,, A,, with corresponding right eigenvectors wl,w2,. . . , w,, and left eigenvectors V I , ut, . . . , u,. If W is the matrix with columns wI, j = 1, . . . , n, V the matrix with columns u,,j = 1, . . . , n, and A the diagonal matrix of eigenvalues, then
A
= WAVT
(3)
For a system of this type, the modes are defined to be the components of the dynamic response of the system referred to the right eigenvectors as basis vectors; they have decay times equal to the reciprocals of the real parts of corresponding eigenvalues. Thus, the free response of the Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
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